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Consider an angle in standard position with r = 12 centimeters, as shown in the figure. Write a short paragraph describing the changes in the magnitudes of x, y, sin Θ, cos Θ, and tan Θ as Θ increases continually from 0° to 90°.

Consider an angle in standard position with r = 12 centimeters, as shown in the figure. Write a short paragraph describing the changes in the magnitudes of x, y class=

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wolffiegirl

x decreases from 12 to 0 
y increases from 0 to 12 
sine increases from 0 to 1 
cosine decreases from 1 to 0 
tangent increases from 0 to infinity

Following are the solution on the given graph:

\(\sin \theta = \frac{y}{12}\\\\\to y= 12 \ \sin \theta \\\\\cos \theta=\frac{x}{12}\\\\\to x= 12 \ \cos \theta \\\\\tan \theta= \frac{y}{x}\)

  • As \(\theta\) increases steadily from \(0^{\circ} \ to \ 90^{\circ},\) while \(\sin \theta\) increases continuously from 0 to 1.
  • As a result, y increases continuously between \(0 (12 \sin (0^{\circ}) \ to\ 12 (12 \sin (90^{\circ}).\)
  • When \(\cos \theta\) declines form 1 to 0, x drops from \(12(12 \cos(0^{\circ})) \ to \ 0 (12 \cos (90^{\circ}).\)
  • \(\tan \theta\), but at the other hand, increases indefinitely beyond 0 and is not defined when \(\theta = 90^{\circ}\).

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